### Financial

### Institutions

### Center

*The Valuation of Default Risk*

*in Corporate Bonds and Interest*

*Rate Swaps*

by

Soren S. Nielsen Ehud I. Ronn 96-23

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Soren S. Nielsen is at the Department of Management Science and Information Systems of the 1

University of Texas at Austin. Ehud I. Ronn is at the Department of Finance of the University of Texas at Austin. Correspondence should be addressed to Ehud I. Ronn, Department of Finance, College and Graduate School of Business, University of Texas at Austin, Austin, TX 78712-1179. Phone: (512) 471-5853. Fax: (512) 471-5073. E-mail: eronn@mail.utexas.edu.

### The Valuation of Default Risk in Corporate Bonds and Interest Rate Swaps

1### October 1994

### Revised: January 15, 1996

### Abstract : This paper implements a model for the valuation of the default risk implicit in

### the prices of corporate bonds. The analytical approach considers the two essential

### ingredients in the valuation of corporate bonds: interest rate uncertainty and default risk.

### The former is modeled as a diffusion process. The latter is modeled as a spread following a

### diffusion process, with the magnitude of this spread impacting on the probability of a

### Poisson process governing the arrival of the default event. We apply two variants of this

### model to the valuation of fixed-for-floating swaps. In the first, the swap is default-free, and

### the spread represents the appropriate discounted expected value of the instantaneous TED

### spread; in the second, we allow the swap to incorporate default risk. We propose to test our

### models using the entire term structure of corporate bonds prices for different ratings and

### industry categories, as well as the term structure of fixed-for-floating swaps.

**The Valuation of Default Risk**

**in Corporate Bonds and Interest Rate Swaps***

**1 Introduction**

The valuation of securities subject to default risk has been of interest in the academic and
practitioner literature for some time. Beginning with Merton (1974), a substantial body
of literature has modeled the default event as dependent on the value of the firm, *F, *with

prespecified default boundary. For the latter, see, e.g., the recent work of Hull and White (1992) and Longstaff and Schwartz (1994). In contrast, the approach we offer below is silent on the causality of default, and hence does not require specification of the precise events which cause the firm to seek, or be forced into, a state of bankruptcy. Hence, this approach enjoys a greater generality relative to its predecessors. Following Jarrow and Turnbull (1992), Jarrow, Lando and Turnbull (1994), Lando (1994), Madan and Unal (1994) and Duffie and Singleton (1994), the current model abstracts from an explicit dependence on an unobservable firm value. In contrast, our model explicitly models the instantaneous default spread on risky bonds, correlates that spread to the riskfree rate of interest, and links the default intensity to its presumed market-efficient predictor, the prevailing instantaneous credit spread.

* The authors acknowledge the helpful comments and suggestions of George Handjinicolaou, Guillaume Fonkenell, Martin Fridson, Dilip Madan, George Morgan and finance seminar partici-pants at the University of Maryland, the University of Texas at Austin and Southern Methodist University. Earlier drafts of this paper were presented at the April 1994 Derivatives and Systemic Risk Conference, the April 1995 Cornell University/Queen’s University Derivative Conference, and the May 1995 Conference of the Society for Computational Economics. The authors would also like to thank Merrill Lynch & Co. for providing data in support of this project. The authors remain solely responsible for any errors contained herein.

The approach we suggest leads to directly testable results in the valuation of corporate bonds. We submit our model to rigorous econometric testing using a data base com-posed of the entire term structures of corporate bonds for different ratings and industry classifications.

Practitioners have of late become interested in models of default risk for several reasons. First, understanding the default event permits a more accurate modeling of issues subject to default in general, and corporate bonds in particular. Second, the modeling of the default event permits the construction and valuation of alternative securities whose payoff is triggered by the default event (or lack thereof), which go by the generic name “credit derivatives.”1

The valuation of interest rate swaps has attracted significant attention. One strand of
literature, including Evans and Bales (1991), Litzenberger (1992), and Chen and Selender
(1994) have reported on the relative *insensitivity *of swap spreads to default measures,
while Grinblatt (1995) has modeled the swap spread as a compensation for illiquidity
relative to Treasuries. In contrast, the works of Cooper and Mello (1991) and Longstaff
and Schwartz (1995) have attributed swap spreads to default risk, while Bhasin (1995) has
recently reported on the increasing cross-sectional credit-quality variety of OTC-derivative

or where

counterparties. Swap contracts are natural candidates for two-factor modeling, and in this paper we present two such models: in the first, the swap spread is simply the riskfree discounted expected present value of the TED (LIBOR — Treasury) spread, whereas the second model generalizes the first to admit default risk.

The paper is organized as follows. Section 2 proposes the two-factor corporate-bond model. Section 3 discusses the data and methodology, with Section 4 reporting on the model’s empirical results. Section 5 presents default-free as well as risky two-factor swaps models, and their attendant empirical estimates. Section 6 concludes.

**2**

**A Two-Factor Model of Default Risk**

**2.1 Model**

The two-factor model of the default risk assumes the following process:

where

(1)

instantaneous default spread *s *and the after-default recovery rate *D. *Specifically, given our
notation, consider the par value of an instantaneously-maturing risky debt issue
promis-valuation, this value is equal to the discounted expected value of the payoffs:

The intuition for this model incorporates the following aspects:

1. The corporation’s bond prices are presumed to drift downward as default becomes increasingly likely (e.g., as they are downgraded).

2. Given the default event has occurred, the bond sells at a post-default price of *D,*

some fraction of face value.3

3. We allow for a correlation in changes between *r *and *s, *and perform period-by-period
discounting using riskless rates of interest.

4. Over the next interval, if the instantaneously-maturing security does not default, it
will pay a return in excess of the riskfree rate; otherwise, it will sell at a price *D.*

**2.2**

**Implementation**

equations (1) specify joint lognormal diffusions — we require a lattice implementation:

3_{In a more general model, the fraction }_{D }_{would not be a constant but presumably depend on}

the prevailing term structure of interest rates. 4

process is:4

(2)

The system (2) results in an efficient, recombining tree; see Appendix A for details.

ance and covariance conditions implied in equations (1).5 _{Specifically, we require that,}

under the risk-neutral probability measure,

(3)

(4) (5) Appendix A demonstrates that these conditions give rise to 10 equations in 12 variables,

4_{The quadrinomial lattice contains the minimal number of nodes required to satisfy the }

first-and second-moment restrictions specified in eqs. (1).

and hence is underidentified. That appendix also demonstrates that the quadrinomial tree

This choice of parameters satisfies the constraints that all probabilities be positive so long Note that the above tree is not a true quadrinomial, for whenever a default occurs, that branch of the tree terminates. In fact, for the postulated constant u’s, we obtain a simple recombining tree.7

**3**

**Data: Term Structures of Treasury Bonds, **

**Corpo-rate Bonds and Swaps**

The data for our analysis consists of the term structures of interest rates for alternate-categories ratings Merrill Lynch & Co. bond indices, specified in Appendix B: these bond indices constitute examples wherein we observe an entire term structure of Treasury and corporate credit spreads through a 10-year maturity; with semi-annually separated inter-vals, this implies data on 20 par bonds with maturities ranging from one-half year to ten years. An interesting component of this data is the swap curve, which in turn permits the application of this methodology to the calculation of the default risk implicit in these important securities.

Our data includes observations on the par rates for Treasury and corporate bonds — including standard fixed-for-floating-LIBOR swap rates — at maturities of two, three,

respectively. From this data, we can extract the zero-coupon Treasury and corporate rates

A, which imply fewer restrictions on parameter values in order to assure positive state prices; the resulting empirical magnitudes did not change substantially.

step of the tree.

through a “bootstrap” technique which complements the observable bonds by linearly interpolating the par rates for each semi-annually separated maturity from six months through ten years. The technique uses the postulated twenty par rates and sequentially solves for the zero-coupon rate that would give rise to the observable par rates. The analytical details of this technique are described in Appendix C.

Figure 1 displays an example of the Treasury and corporate par rate curve for several ratings categories, including the swap curve.

**4 Default-Risk Parameter Estimation for Corporate**

**Bonds**

**4.1 Methodology**

Using this data, our analysis determines the *implicit *values of the parameters which
explain the cross-sectional term structures of interest rates for alternate ratings categories.
Specifically, assume that we observe the term structure of the riskless rate of interest,
and the term structure of credit spreads for a given ratings category. This data is conveyed

the market prices of the Treasury corporate securities, respectively, for maturity t, t = 0.5, 1, . . . ,

8_{The valuation of swaps is perforce different, since the widening/tightening of the swap spread}

does not necessarily imply a concomitant increase/decrease in the probability of a swap counter-value of the swap at a given point in time.

process for the default spread is then time-homogeneous and entirely determined by the

Ideally, one would implement an empirical procedure to estimate the value of the

pa-is immediate default, but with full recovery. While thpa-is pa-is a feasible empirical solution, it pa-is
clearly an unintuitive one. In order to preclude such a solution, we fix *D *at two arbitrary
values, *D *= 0 and *D = *.5.

distributed, then the estimation problem is

we have 20 data points and three parameters to estimate.

In calibration estimations such as these, we infrequently encounter the possibility of estimating an implied correlation coefficient. In the current case, objective (6) permits the

The estimation procedure can be ratings category-specific or it can be market-wide: ries. Thus, the objective function (6) could be estimated with and without imposing the identical parameter values across different ratings categories.9

**4.2**

**Statistical Properties of the Estimates**

The above model is amenable to non-linear least squares estimation. If

where

=

**4.3 Empirical Results**

For 12/29/94, Table 1 reports the following set of parameter estimates:

Table 1 — Parameter Estimates for 12/29/94

Note: Numbers in parentheses represent standard errors.

For *D *= 0, Panel A of Table 1 reports the separate-category estimates of implied
vola-on 12/29/94 as equal to {.130, .115, .862}.
Figure 2 contains a perspective on these values by presenting the time-series estimates of
these parameters for the entire month of December 1994.

For *D *= 0.5, Figure 3 displays the time-series of implied volatility for the riskless rate
remarkable stability for the fourth quarter of 1994. Similar results also obtain for *D = *0.
We performed sensitivity analyses on the parameter estimates and observed the

ing effects:

Interestingly, the impact of the correlation effect was negative, but only weakly so. Our model’s empirical result thus contrasts with the finding of Longstaff and Schwartz (1994), who find that “the correlation of a firm’s assets with changes in the level of the interest rate can have significant effects on the value of risky fixed-income securities” (p. 23).

**5**

**A Two-Factor Model for the Swap Curve**

**5.1**

**A Two-Factor Default-Free Model for the Swap Curve**

As noted above, swap contracts are natural candidates for two-factor modeling. Thus, our first approach models the swap spread as the riskfree discounted expected present value of the TED spread.

to-market value of an interest rate swap receiving fixed and paying floating LIBOR on a

Now consider the valuation of put swaptions.

notional amount) of a put swaption with time to expiration *t, *swap-maturity *T *and exercise
rate K. The value of the put swaption is the discounted expected value of the payoff,

put swaptions prices *P (t, T, K),* we can obtain empirical estimates of the model’s

para-then the estimation procedure is

order to place emphasis on the higher-order volatility-dependence of swaption prices.

**5.2**

**A Two-Factor Risk Model for the Swap Curve**

A default-risk approach to the valuation of the swap curve requires a modification of the corporate bond-default model. Specifically, if the swap is the sole contract existing between the two parties, and the two parties currently have identical credit rating, then the probability of default increases with the absolute value of the mark-to-market on the swap. We require this assumption of default symmetry, since our empirical swap rates mark-to-market value of an interest rate swap receiving fixed and paying floating LIBOR

10_{For a modeling of the swap contract with asymmetric default assumptions, see Duffie and}

Huang (1995).

partial-recovery-rate parameter *k < *1,

(7) where

occurring over the next interval11

specified in the system (2).

assume that in the region of interest, we can write the linear relationship

In this case,

the values of the put swaptions are once again since the short side of the put swaption has no incentive to default prior to the exercise of the put. However, the arguments of the value

**5.3 Empirical Results**

Using the above data, the estimation problem is now

of the model’s coefficients, as well as the test of whether the incorporation of default risk

residuals, the relevant test statistic for the nested test becomes

Table 2 reports the empirical tests based on vanilla fixed-for-floating swap rates out to a ten year maturity, and 17 put swaption prices for each of eight dates over the period Jan. 18, 1993 — May 3, 1994.

Several results from Table 2 are noteworthy:

1. At the 5% level, the sum of squared residuals on each observation date is significantly reduced through the addition of the possibility of default. If we posit the identical parameter values throughout the sample period, then the significance of default risk can be established at the 1% level.

of riskfree rates of interest.

**6 S u m m a r y**

This paper has derived a two-factor risk model for corporate bonds and interest rate swaps. The model is parsimonious in its assumptions, uses readily-observable inputs and is esti-mated empirically from the observable spreads of defaultable bonds and fixed-for-floating interest rate swaps. The empirical results suggest the importance of explicit modeling of the default event for both sets of securities. Further, we believe subsequent work should be focused on modeling the relevant stochastic processes so as to elicit the implied after-default values of corporate bonds and interest-rate swaps.

**A**

**Construction of the Quadrinomial Tree**

Equation (2) in the text specifies the quadrinomial lattice to be used for corporate bond valuation. This appendix demonstrates the numerical validity of that specification in fulfilling the first and second moment requirements of the joint process eqs. (1).

Consider first the first-moment conditions (3). For *i *= 1, 2, we require that

(8)

(9) to produce the following two conditions:

which implies

Finally, the covariance condition (5) is

which implies

We now demonstrate the tree can be constructed to satisfy all requirements under the

equations in six unknowns:

(10) (11) (12) (13) (14) (15) The solution to this system can be seen by inspection as follows. Equations (11) and

the solution to the system is

The non-uniqueness of this system can be resolved by appeal to the economics of the 19

**B D a t a**

The data consists of the term structures of interest rates for alternate ratings-categories Merrill Lynch & Co. bond indices.

**C**

**“Bootstrapping” the Yield Curve from Par Rates**

For a unit par bond, consider the relation

(16)

(17)

( 1 8 ) From eq. (18),

on-the-run points 2, 3, 7 and 10 yrs.

**References**

1. Bhasin, Vijay, “On the Credit Risk of OTC Derivative Users,” Working Paper, Fed-eral Reserve Board of Governors, 1995.

2. Chen, Andrew and Arthur K. Selender, “Determination of Swap Spreads: An Em-pirical Analysis,” Working Paper, Southern Methodist University, 1994.

3. Cooper, Ian and Antonio Mello, “The Default Risk of Swaps,” *Journal of Finance,*

46, 1991, pp. 597 - 620. item Duffie, Darrell and Ming Huang, “Swap Rates and Credit Quality,” Working Paper, Stanford University, 1995.

4. Evans, E. and Giola Parente Bales, “What Drives Interest Rate Swap Spreads?” in Interest Rate Swaps, Carl Beidleman (ed.), 1991, pp. 280 - 303.

5. Grinblatt, Mark, “An Analytic Solution for Interest Rate Swap Spreads,” Working Paper, Anderson Graduate School of Management, April 1995.

6. Hull, John and Alan White, “The Impact of Default Risk on the Prices of Options and Other Derivative Securities,” Working Paper, Faculty of Management, University of Toronto, 1992.

7. Jarrow, Robert A. and Stuart Turnbull, “Pricing Options on Financial Securities Subject to Credit Risk,” Working Paper, Graduate School of Management, Cornell University, 1992.

8. Jarrow, Robert A., David Lando and Stuart Turnbull, “A Markov Model for the Term Structure of Credit Risk Spreads,” Working Paper, Cornell University and Queen’s University, 1994.

9. Lando, David, “A Continuous-Time Markov Model of the Term Structure of Credit Spreads,” Working Paper, Graduate School of Management, Cornell University, 1993.

10. Leland, Hayne E. and Klaus B. Toft, “Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads,” Working Paper, Univer-sity of California, Berkeley, December 1995.

11. Litzenberger, Robert H., “Swaps: Plain and Fanciful,” *Journal of Finance, *1992, pp.
597 - 620.

12. Longstaff, Francis and Eduardo Schwartz, “A Simple Approach to Valuing Risky
Fixed and Floating Rate Debt,” *Journal of Finance, *50, pp. 789 - 821.

13. Madan, Dilip B. and Haluk Unal, “Pricing the Risks of Default,” Working Paper, College of Business, University of Maryland, 1994.

14. Merton, Robert C., “On the Pricing of Corporate Debt: The Risk Structure of
Interest Rates,” *Journal of Finance, *1974, pp. 449-470.